Studying an underground zone requires construction of numerical reservoir models. A numerical model consists of a grid with N dimensions (N>0 and generally two or three) each cell of which is assigned the value of a property characteristic of the zone studied. It can be, for example, the porosity or the permeability distributed in a reservoir. Such a value is referred to as regionalized variable. It is a continuous variable, spatially distributed, and representative of a physical phenomenon. From a mathematical point of view, it is simply a function z(u) taking a value at each point u (the cell of the grid) of a field of study D (the grid representative of the reservoir). However, the variation of the regionalized variable in this space is too irregular to be formalized by a mathematical equation. In fact, the regionalized variable represented by z(u) has both a global aspect relative to the spatial structure of the phenomenon studied and a random local aspect.
This random local aspect can be modelled by a random variable (VA). A random variable is a variable that can take a certain number of realizations z according to a probability law. Continuous variables such as seismic attributes (acoustic impedance) or petrophysical properties (saturation, porosity, permeability) can be modelled by VAs. Therefore, at point u, the regionalized variable z(u) can be considered to be the realization of a random variable Z.
However, to properly represent the spatial variability of the regionalized variable, it must be possible to take into account the double aspect, both random and structured. One possible approach, of probabilistic type, involves the notion of random function. A random function (FA) is a set of random variables (VA) defined in a field of study D (the grid representative of the reservoir), i.e. {Z(u), u∈D}, also denoted by Z(u). Thus, any group of sampled values {z(u1), . . . ,z(un)} can be considered to be a particular realization of random function Z(u)={Z(u1), . . . ,Z(un)}. Random function Z(u) allows to take into account both the locally random aspect (at uα, the regionalized variable z(uα) being a random variable) and the structured aspect (via the spatial probability law associated with random function Z(u)).
The realizations of a random function provide stochastic reservoir models. From such models, it is possible to appreciate the way the underground zone studied works. For example, simulation of the flows in a porous medium represented by numerical stochastic models allows, among other things, to predict the reservoir production and thus to optimize its development by testing various scenarii.
Construction of a stochastic reservoir model can be described as follows:
First, static data are measured in the field on the one hand (logging, measurements on samples taken in wells, seismic surveys, . . . ) and, on the other hand, dynamic data are measured (production data, well tests, breakthrough time, . . . ), whose distinctive feature is that they vary in the course of time as a function of fluid flows in the reservoir,
then, from the static data, a random function characterized by its covariance function (or similarly by its variogram), its variance and its mean is defined,
besides, a set of random numbers drawn independently of one another is defined: it can be, for example, a Gaussian white noise or uniform numbers. We thus have an independent random number per cell and per realization,
finally, from a selected geostatistical simulator and from the set of random numbers, a random draw in the random function is performed, giving access to a (continuous or discrete) realization representing a possible image of the reservoir. In general, this realization does not verify the punctual measurements. An additional operation based on kriging techniques allows to modify the realization by making it meet these measurements. This technique is described for example in the following document:
Chilès, J.P., and Delfiner, P., 1999, Geostatistics—Modeling spatial uncertainty, Wiley series in probability and statistics, New York, USA,
at this stage, the dynamic data have not been considered. They are integrated in the reservoir models via an optimization or a calibration. An objective function measuring the difference between the dynamic data measured in the field and the corresponding responses simulated for the model considered is defined. The goal of the optimization procedure is to modify little by little this model so as to reduce the objective function.
In the end, the modified models are coherent in relation to the static data and the dynamic data. This reservoir model construction stage requires considerable engineering time and calculating time.
Since the reservoir model calibration techniques using dynamic data have markedly improved during the past years as a result of the power increase of computers and of the appearance of new parametrization techniques, and since the hydrocarbon recovery techniques have also markedly improved, reservoir engineers frequently need to go back over reservoir models worked out in the past. The goal is to fine down these models and to update them by means of the data acquired since the time when the model had been initially worked out.
However, an essential difficulty still remains when going back over numerical models worked out in the past. In fact, to apply a method allowing to fine down calibration of an existing realization, the number of random numbers which, when given to the geostatistical simulator, provides the numerical model (the realization) in question has to be known. Now, in general, this information no longer exists. Similarly, the variogram (or covariance) model characterizing the spatial variability in the underground zone of the attribute represented and necessary to characterize the random function is no longer known. The latter point is less important insofar as a study of the existing numerical model can allow to find this variogram again. On the other hand, there is so far no method for determining the set of random numbers underlying the numerical model.
The method according to the invention allows to reconstruct numerical stochastic models, i.e., for a previously determined random function, to identify a set of random numbers which, input into a geostatistical simulator, leads to a realization similar to the numerical model considered. It comprises several algorithms, iterative or not, which, depending on the conditions, allow to reconstruct continuous or discrete realizations.
Furthermore, the reconstruction techniques developed are compatible with the gradual deformation method, a geostatistical parametrization technique meeting engineers' needs and presented in the following article:
Hu, L.-Y., 2000, Gradual deformation and iterative calibration of Gaussian-related stochastic models, Math. Geol., 32(1), 87-108.
In fact, it allows to resume the study of a reservoir and to modify it partly in the zones where new data are available, as described in the document as follows:
Le Ravalec-Dupin, M., Noetinger, B., Hu, L.-Y., and Blanc, G., 2001, Conditioning to dynamic data: an improved zonation approach, Petroleum Geosciences, 7, S9-S16.